cdlemg10a - Mathbox for Norm Megill

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Theorem cdlemg10a 40623

Description: TODO: FIX COMMENT. (Contributed by NM, 3-May-2013.)

**Hypotheses**
Ref	Expression
cdlemg8.l	⊢ ≤ = (le‘𝐾)
cdlemg8.j	⊢ ∨ = (join‘𝐾)
cdlemg8.m	⊢ ∧ = (meet‘𝐾)
cdlemg8.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg8.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg8.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg10.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
cdlemg10a	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) ≤ ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))

**Proof of Theorem cdlemg10a**
Step	Hyp	Ref	Expression
1		simp11 1202	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp12 1203	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
3		simp13 1204	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
4		simp21 1205	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝐹 ∈ 𝑇)
5		simp22 1206	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝐺 ∈ 𝑇)
6		simp23 1207	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝑃 ≠ 𝑄)
7		simp31 1208	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))
8		cdlemg8.l	. . . 4 ⊢ ≤ = (le‘𝐾)
9		cdlemg8.j	. . . 4 ⊢ ∨ = (join‘𝐾)
10		cdlemg8.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
11		cdlemg8.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
12		cdlemg8.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
13		cdlemg8.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
14	8, 9, 10, 11, 12, 13	cdlemg9 40617	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) ≤ ((((𝐹‘(𝐺‘𝑃)) ∨ (𝐺‘𝑃)) ∧ ((𝐹‘(𝐺‘𝑄)) ∨ (𝐺‘𝑄))) ∨ (((𝐺‘𝑃) ∨ 𝑃) ∧ ((𝐺‘𝑄) ∨ 𝑄))))
15	1, 2, 3, 4, 5, 6, 7, 14	syl133anc 1392	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) ≤ ((((𝐹‘(𝐺‘𝑃)) ∨ (𝐺‘𝑃)) ∧ ((𝐹‘(𝐺‘𝑄)) ∨ (𝐺‘𝑄))) ∨ (((𝐺‘𝑃) ∨ 𝑃) ∧ ((𝐺‘𝑄) ∨ 𝑄))))
16	8, 11, 12, 13	ltrnel 40122	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊))
17	1, 5, 2, 16	syl3anc 1370	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊))
18	8, 11, 12, 13	ltrnel 40122	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝐺‘𝑄) ∈ 𝐴 ∧ ¬ (𝐺‘𝑄) ≤ 𝑊))
19	1, 5, 3, 18	syl3anc 1370	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝐺‘𝑄) ∈ 𝐴 ∧ ¬ (𝐺‘𝑄) ≤ 𝑊))
20		simp12l 1285	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ 𝐴)
21		simp13l 1287	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝑄 ∈ 𝐴)
22	11, 12, 13	ltrn11at 40130	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → (𝐺‘𝑃) ≠ (𝐺‘𝑄))
23	1, 5, 20, 21, 6, 22	syl113anc 1381	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐺‘𝑃) ≠ (𝐺‘𝑄))
24		simp32 1209	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))
25		cdlemg10.r	. . . . . . . 8 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
26	8, 9, 10, 11, 12, 13, 25	cdlemg10c 40622	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝑅‘𝐹) ≤ ((𝐺‘𝑃) ∨ (𝐺‘𝑄)) ↔ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄)))
27	1, 2, 3, 4, 5, 26	syl122anc 1378	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑅‘𝐹) ≤ ((𝐺‘𝑃) ∨ (𝐺‘𝑄)) ↔ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄)))
28	24, 27	mtbird 325	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ¬ (𝑅‘𝐹) ≤ ((𝐺‘𝑃) ∨ (𝐺‘𝑄)))
29	8, 9, 10, 11, 12, 13, 25	trlval4 40171	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊) ∧ ((𝐺‘𝑄) ∈ 𝐴 ∧ ¬ (𝐺‘𝑄) ≤ 𝑊)) ∧ ((𝐺‘𝑃) ≠ (𝐺‘𝑄) ∧ ¬ (𝑅‘𝐹) ≤ ((𝐺‘𝑃) ∨ (𝐺‘𝑄)))) → (𝑅‘𝐹) = (((𝐺‘𝑃) ∨ (𝐹‘(𝐺‘𝑃))) ∧ ((𝐺‘𝑄) ∨ (𝐹‘(𝐺‘𝑄)))))
30	1, 4, 17, 19, 23, 28, 29	syl132anc 1387	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑅‘𝐹) = (((𝐺‘𝑃) ∨ (𝐹‘(𝐺‘𝑃))) ∧ ((𝐺‘𝑄) ∨ (𝐹‘(𝐺‘𝑄)))))
31		simp11l 1283	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ HL)
32	8, 11, 12, 13	ltrnat 40123	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐺‘𝑃) ∈ 𝐴)
33	1, 5, 20, 32	syl3anc 1370	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐺‘𝑃) ∈ 𝐴)
34	8, 11, 12, 13	ltrnat 40123	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝐺‘𝑃) ∈ 𝐴) → (𝐹‘(𝐺‘𝑃)) ∈ 𝐴)
35	1, 4, 33, 34	syl3anc 1370	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐹‘(𝐺‘𝑃)) ∈ 𝐴)
36	9, 11	hlatjcom 39350	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝐺‘𝑃) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑃)) ∈ 𝐴) → ((𝐺‘𝑃) ∨ (𝐹‘(𝐺‘𝑃))) = ((𝐹‘(𝐺‘𝑃)) ∨ (𝐺‘𝑃)))
37	31, 33, 35, 36	syl3anc 1370	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝐺‘𝑃) ∨ (𝐹‘(𝐺‘𝑃))) = ((𝐹‘(𝐺‘𝑃)) ∨ (𝐺‘𝑃)))
38	8, 11, 12, 13	ltrnat 40123	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴) → (𝐺‘𝑄) ∈ 𝐴)
39	1, 5, 21, 38	syl3anc 1370	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐺‘𝑄) ∈ 𝐴)
40	8, 11, 12, 13	ltrnat 40123	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝐺‘𝑄) ∈ 𝐴) → (𝐹‘(𝐺‘𝑄)) ∈ 𝐴)
41	1, 4, 39, 40	syl3anc 1370	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐹‘(𝐺‘𝑄)) ∈ 𝐴)
42	9, 11	hlatjcom 39350	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝐺‘𝑄) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑄)) ∈ 𝐴) → ((𝐺‘𝑄) ∨ (𝐹‘(𝐺‘𝑄))) = ((𝐹‘(𝐺‘𝑄)) ∨ (𝐺‘𝑄)))
43	31, 39, 41, 42	syl3anc 1370	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝐺‘𝑄) ∨ (𝐹‘(𝐺‘𝑄))) = ((𝐹‘(𝐺‘𝑄)) ∨ (𝐺‘𝑄)))
44	37, 43	oveq12d 7449	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (((𝐺‘𝑃) ∨ (𝐹‘(𝐺‘𝑃))) ∧ ((𝐺‘𝑄) ∨ (𝐹‘(𝐺‘𝑄)))) = (((𝐹‘(𝐺‘𝑃)) ∨ (𝐺‘𝑃)) ∧ ((𝐹‘(𝐺‘𝑄)) ∨ (𝐺‘𝑄))))
45	30, 44	eqtrd 2775	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑅‘𝐹) = (((𝐹‘(𝐺‘𝑃)) ∨ (𝐺‘𝑃)) ∧ ((𝐹‘(𝐺‘𝑄)) ∨ (𝐺‘𝑄))))
46		simp33 1210	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))
47	8, 9, 10, 11, 12, 13, 25	trlval4 40171	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑅‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃)) ∧ (𝑄 ∨ (𝐺‘𝑄))))
48	1, 5, 2, 3, 6, 46, 47	syl132anc 1387	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑅‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃)) ∧ (𝑄 ∨ (𝐺‘𝑄))))
49	9, 11	hlatjcom 39350	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐺‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐺‘𝑃)) = ((𝐺‘𝑃) ∨ 𝑃))
50	31, 20, 33, 49	syl3anc 1370	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ (𝐺‘𝑃)) = ((𝐺‘𝑃) ∨ 𝑃))
51	9, 11	hlatjcom 39350	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝐺‘𝑄) ∈ 𝐴) → (𝑄 ∨ (𝐺‘𝑄)) = ((𝐺‘𝑄) ∨ 𝑄))
52	31, 21, 39, 51	syl3anc 1370	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑄 ∨ (𝐺‘𝑄)) = ((𝐺‘𝑄) ∨ 𝑄))
53	50, 52	oveq12d 7449	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐺‘𝑃)) ∧ (𝑄 ∨ (𝐺‘𝑄))) = (((𝐺‘𝑃) ∨ 𝑃) ∧ ((𝐺‘𝑄) ∨ 𝑄)))
54	48, 53	eqtrd 2775	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑅‘𝐺) = (((𝐺‘𝑃) ∨ 𝑃) ∧ ((𝐺‘𝑄) ∨ 𝑄)))
55	45, 54	oveq12d 7449	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) = ((((𝐹‘(𝐺‘𝑃)) ∨ (𝐺‘𝑃)) ∧ ((𝐹‘(𝐺‘𝑄)) ∨ (𝐺‘𝑄))) ∨ (((𝐺‘𝑃) ∨ 𝑃) ∧ ((𝐺‘𝑄) ∨ 𝑄))))
56	15, 55	breqtrrd 5176	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) ≤ ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 lecple 17305 joincjn 18369 meetcmee 18370 Atomscatm 39245 HLchlt 39332 LHypclh 39967 LTrncltrn 40084 trLctrl 40141

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-riotaBAD 38935

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-undef 8297 df-map 8867 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142

This theorem is referenced by: cdlemg10 40624

W3C validator